18 CANONICAL DECOMPOSITIONS OF A MEASURE 211 to obtain an increasing sequence of sets Am <= X such that ^(X — Am) -> 0 as m -> oo n and in which the partial sums sn(x) =]T \fk(x)\2 tend uniformly to +00. Then remark fc = i that the series with general term \fn(x)\2lsn(x) is divergent almost everywhere (Section 5.3, Problem 6). (c) Under the same hypotheses show that for almost all XQ e X there exists a function g e ^c(X, p,) (depending on x0) such that the series with general term (g \fn)fn(xQ) has sum equal to 4- co (use Problem 23 of Section 12.16). 6. (a) Let 6 : R->R be increasing and continuous on the right. Show that there exists a unique positive measure v on R such that v(\a, b])=6(b)~ 6(a) for every half-open interval ]a,b]. (This measure is called the "Stieltjes measure defined by 6", and we write fdB in place of fdv.) Conversely, every positive measure on R may be obtained in this way, and two functions 0Z, 02 on R (both of them increasing and continuous on the right) define the same measure if and only if 92 ~~ &i is constant. Under what conditions is the measure v diffuse? What is then the image of v under 0? (b) Let K be the Cantor set (Section 4.2, Problem 2). Show that there exists a diffuse positive measure v on R with support K and total mass equal to 1 (Section 4.2, Problem 2(d)). Deduce that there exists on R a diffuse positive measure, disjoint from Lebesgue measure, with support equal to I = [0, 1 ]. (In each component interval J of I — K choose a measure proportional to the image of v under an affine linear mapping of I onto J; then proceed by induction.) (c) Deduce from (a) that if ^ is a positive measure on R+ and/a function belonging to &ioc, c(R + » /*)» anc* (O a dense sequence in R such that fdu = 0 for all n, J[0, tn] then/is /z-negligible. 7. Let X be a compact space and /x a diffuse positive measure on X with total mass equal to 1. (a) Construct a family (U(O)o^r^i of /^-quadrable open sets, such that U(0) ~ 0, U(l) = X, U(0 c U(O whenever / < /', and j*(U(0)« * for all t. (First define the U(0 for t of the form /c/2", by induction on n. Use Problem 7(d) of Section 13.9 to prove that if V, W are two quadrable open sets in X such that V c: w and /x( V) < jn(VV), then there exists a quadrable open set U such that V c u <= 0 c W and i/i(W - V)< JK(U - V) < J/x(W - V). Also use the result of Problem 3(b).) (b) Deduce from (a) that there exists a continuous mapping TT of X onto I = [0,1] such that the image of /x under TT is Lebesgue measure A on I. (c) Show that there exists a /^-negligible subset N of X, a A-negligible subset M of I and a homeomorphism TTO of I — M onto X — N such that, for every A-measurable subset A of I — M, the set TTO(A) is /x-measurable and of measure JJL(TTQ(A)) — A(A). (Use Problem 7(d) of Section 13.9 to show that for each integer n> 0 there exists a finite partition of X consisting of a /^-negligible set and quadrable open sets of diameter <l/n (with respect to a distance defining the topology of X) and of measure gl/w. Proceeding by induction on n and passing to the limit, obtain a homeomorphism/ of I — D onto a subset of X of measure 1, where D is a denumerable subset of I.) 8. Let X, Y be two compact spaces, U a continuous linear mapping of the Banach space tfR(X) into the Banach space #n(Y), and 7r:X->Y a continuous mapping. fc=l